Magic math for the meandering mind

If meetings and wedding receptions bore you, maybe you would enjoy magic squares. A magic square is a square divided into smaller squares, its columns, rows, and main diagonals all adding up to the same number. Magic squares were popular in 18th-century Europe, especially in France, where entire books were devoted to them, having come from India and China, where they were known since before Christ’s time.
Benjamin Franklin studied magic squares in his youth and excelled at them, as he did in any subject to which he applied himself, and his biographer Carl Van Doren wrote, “His mind, growing tired of business … swung in inquisitive directions. Though it was patient, it seems never to have known fatigue or languor. Franklin considered squares that totaled the same number vertically, horizontally and diagonally common and easy things. In later years he told a visitor, ‘I had amused myself in making these kind of magic squares, and at length had acquired such a knack at it that I could fill the cells of any magic square, of reasonable size, with a series of numbers as fast as I could write them’ …. His visitor showed him a book on magic squares and commented on a 256-cell square (16 cells to a side) that must have been a work of great labor, so Franklin that night made a 256-cell square that in addition to the rows, columns and 16-cell diagonals totaling 260, also had many bent diagonals that totaled 260 and … ‘had this added: that a four-square hole being cut in a piece of paper of such a size as to take in and show through it just 16 of the little squares, when laid on the greater square, the sum of the 16 numbers so appearing through the hole, wherever it was placed on the greater square, should likewise make 2056,’ wrote Franklin.” Van Doren said that a Frenchman, Barbeu Dubourg, when translating Franklin’s works, found two mistakes in the square.
Many methods were developed to make magic squares, but it is difficult to explain them without diagrams. Squares that start with one and proceed by ones through the natural numbers are called primitive magic squares, but you may start with any number you want, and you may use other increments as long as the increments are regular. In other words, you may start, for example, with 10 and add three each time: 10, 13, 16, etc. Learn this method and you can, as Franklin did, fill in the numbers as quickly as you can write them. If you’re like me and you don’t dance at wedding receptions, you can do magic squares on the tablecloth (unless it’s cloth, in which case the owner of the hall would frown on that). Start with some big figure and go in leaps, so Cousin Larry will think you’re entering numbers at random, and he’ll be amazed. He doesn’t dance either, so telling him to add those numbers will give him something to do.
As with Cousin Larry and I, Benjamin Franklin avoided boredom by making magic squares. “Forced to sit idle, as he sometimes was in dragging sessions of the Assembly, he would work out mathematical puzzles,” wrote Van Doren, but Franklin told his visitor that his time spent in his younger days might have been employed more usefully. I disagree. I don’t have much occasion to do math, but I love it, so magic square exercises flex that part of my brain that often lies dormant. As with crossword puzzles, I can feel my brain growing and sharpening when I do magic squares, and I am prepared for situations when I need that type of reasoning.

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